thermodynamics - Can a single classical particle have any entropy?

recently I have had some exchanges with @Marek regarding entropy of a single classical particle.

I always believed that to define entropy one must have some distribution. In Quantum theory, a single particle can have entropy and I can easily understand that. But I never knew that entropy of a single rigid classical particle is a well defined concept as Marek claimed. I still fail to understand that. One can say that in the classical limit, the entropy of a particle in QT can be defined and that corresponds the entropy of a single classical particle. But I have difficulty accepting that that gives entropy of a single Newtonian particle. In my understanding, If a system has entropy then it also should have some temperature. I don't understand how one would assign any temperature to a single classical particle. I came across a paper where there is a notion of "microscopic entropy". By no means, in my limited understanding, it corresponded to the normal concept of entropy. I am curious to know, what is the right answer.

So, my question is, is it possible to define entropy of a single classical particle?

Answer

Entropy is a concept in thermodynamics and statistical physics but its value only becomes indisputable if one can talk in terms of thermodynamics, too.

To do so in statistical physics, one needs to be in the thermodynamic limit i.e. the number of degrees of freedom must be much greater than one. In fact, we can say that the thermodynamic limit requires the entropy to be much greater than one (times $k_B$, if you insist on SI units).

In the thermodynamic limit, the concept of entropy becomes independent of the chosen ensembles - microcanonical vs canonical etc. - up to corrections that are negligible relatively to the overall entropy (either of them).

A single particle, much like any system, may be assigned the entropy of $\ln(N)$ where $N$ is the number of physically distinct but de facto indistinguishable states in which the particle may be. So if the particle is located in a box and its wave function may be written as a combination of $N$ small wave packets occupying appropriately large volumes, the entropy will be $\ln(N)$.

However, the concept of entropy is simply not a high-precision concept for systems away from the thermodynamic limit. Entropy is not a strict function of the "pure state" of the system; if you want to be precise about the value, it also depends on the exact ensemble of the other microstates that you consider indistinguishable.

If you consider larger systems with $N$ particles, the entropy usually scales like $N$, so each particle contributes something comparable to 1 bit to the entropy - if you equally divide the entropy. However, to calculate the actual coefficients, all the conceivable interactions between the particles etc. matter.